At-the-money short-time call-price asymptotics for new classes of exponential L\'evy models

TL;DR

This paper derives at-the-money short-time asymptotics for call prices and implied volatility in exponential Lévy models, revealing how stable laws influence convergence rates and the dominance of Gaussian components.

Contribution

It introduces new short-time asymptotic formulas for exponential Lévy models with stable law attraction, including cases with no Brownian component and models with non-constant slowly varying functions.

Findings

Stable law attraction governs short-time asymptotics.

Jump contributions are lower order when a Brownian component exists.

New convergence rates involving slowly varying functions are established.

Abstract

We develop at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a class of asset-price models whose log returns follow a L\'evy process. Under mild assumptions placing the driving L\'evy process in the small-time domain of attraction of an $\alpha$-stable law with $\alpha \in (1,2)$, we give first-order at-the-money call-price and implied volatility asymptotics. A key observation is that both the stable domain of attraction and the finiteness of the centering constant $\bar{\mu}$ are preserved under the share measure transformation, so that all of the distributional input needed for the call-price expansion can be read off from the regular variation of the L\'evy measure near the origin. When the L\'evy process has no Brownian component, new rates of convergence of the form $t^{1/\alpha} \ell(t)$ where $\ell$ is a slowly varying function are obtained. We provide an example of an exponential L\'evy model exhibiting this behavior, with $\ell$ not asymptotically constant, yielding a convergence rate of $(t / \log(1/t))^{1/\alpha}$. In the case of a L\'evyprocess with Brownian component, we show that the jump contribution is always lower order, so that the leading $\sqrt{t}$ behavior of the at-the-money call price is universal and driven entirely by the Gaussian part of the characteristic triplet.